Hello everyone,

I recently made the grievous mistake of thinking that the elmhes.c routine in NR did not function as intended. I reached this conclusion because when I compared the output of gnu-octave (3.4.2) using the [u,h]=hess(a) routine the results differed from what I got using elmhes on the same matrix.

However, I went ahead and performed the hqr analysis on the hessenberg factorization from elmhes and it identified exactly the same eigenvalues as were reported by octave using the [u,v]=eig(a) call.

This is my question (Sorry for the rather verbose intro): Is it possible to compute two different hessenbergs for the same matrix (whilst preserving the eigenvalues)? Apparently so, but it is not intuitive... Someone who is more knowledgeable than I might offer some enlightenment. For example: when is this not possible?

Thanks,
Mark Allyn

Hi Mark,

The Hessenberg decomposition is not unique. Details are e.g. in Golub and Van Loan's book, section 7.4.5. You can always check two different purported decompositions by seeing that Q H Q^T gives the original matrix A for both cases.

Best,
Saul Teukolsky

Good evening, Saul,

Thank you for confirming my hunch.

Regards,
Mark